Discrete mechanics and variational integrators

TLDR: In this paper, a review of integration algorithms for finite dimensional mechanical systems that are based on discrete variational principles is presented, including the Verlet, SHAKE, RATTLE, Newmark, and the symplectic partitioned Runge-Kutta schemes.

Abstract: This paper gives a review of integration algorithms for finite dimensional mechanical systems that are based on discrete variational principles. The variational technique gives a unified treatment of many symplectic schemes, including those of higher order, as well as a natural treatment of the discrete Noether theorem. The approach also allows us to include forces, dissipation and constraints in a natural way. Amongst the many specific schemes treated as examples, the Verlet, SHAKE, RATTLE, Newmark, and the symplectic partitioned Runge–Kutta schemes are presented.

Figures

Fig. 2. Energy of a dissipative system computed with variational midpoint and Runge–Kutta. Note the accurate energy behaviour of the variational method

Fig. 1. Energy computed with variational Newmark (solid line) and Runge–Kutta (dashed line). Note that the variational method does not artificially dissipate energy

Full text

Acta Numerica (2001), pp. 357–514
c
° Cambridge University Press, 2001
Discrete mechanics and
variational integrators
J. E. Marsden and M. West
Control and Dynamical Systems 107-81,
Caltech, Pasadena, CA 91125-8100, USA
E-mail: marsden@cds.caltech.edu
mwest@cds.caltech.edu
This paper gives a review of integration algorithms for finite dimensional
mechanical systems that are based on discrete variational principles. The
variational technique gives a unified treatment of many symplectic schemes,
including those of higher order, as well as a natural treatment of the discrete
Noether theorem. The approach also allows us to include forces, dissipation
and constraints in a natural way. Amongst the many specific schemes treated
as examples, the Verlet, SHAKE, RATTLE, Newmark, and the symplectic
partitioned Runge–Kutta schemes are presented.
CONTENTS
PART 1: Discrete variational mechanics
1.1 Introduction 359
1.2 Background: Lagrangian mechanics 365
1.3 Discrete variational mechanics:
Lagrangian viewpoint 370
1.4 Background: Hamiltonian mechanics 376
1.5 Discrete variational mechanics:
Hamiltonian viewpoint 383
1.6 Correspondence between discrete and
continuous mechanics 386
1.7 Background: Hamilton–Jacobi theory 390
1.8 Discrete variational mechanics:
Hamilton–Jacobi viewpoint 392

358 J. E. Marsden and M. West
PART 2: Variational integrators
2.1 Introduction 394
2.2 Background: Error analysis 397
2.3 Variational error analysis 399
2.4 The adjoint of a method and symmetric methods 402
2.5 Composition methods 405
2.6 Examples of variational integrators 408
PART 3: Forcing and constraints
3.1 Background: Forced systems 421
3.2 Discrete variational mechanics with forces 423
3.3 Background: Constrained systems 430
3.4 Discrete variational mechanics with constraints 438
3.5 Constrained variational integrators 444
3.6 Background: Forced and constrained systems 452
3.7 Discrete variational mechanics with forces
and constraints 456
PART 4: Time-dependent mechanics
4.1 Introduction 463
4.2 Background: Extended Lagrangian mechanics 464
4.3 Discrete variational mechanics:
Lagrangian viewpoint 472
4.4 Background: Extended Hamiltonian mechanics 480
4.5 Discrete variational mechanics:
Hamiltonian viewpoint 486
4.6 Correspondence between discrete and
continuous mechanics 490
4.7 Background: Extended Hamilton–Jacobi theory 494
4.8 Discrete variational mechanics:
Hamilton–Jacobi viewpoint 496
4.9 Time-dependent variational integrators 497
PART 5: Further topics
5.1 Discrete symmetry reduction 503
5.2 Multisymplectic integrators for PDEs 503
5.3 Open problems 504
References 507

Discrete mechanics and variational integrators 359
PART ONE
Discrete variational mechanics
1.1. Introduction
This paper gives a survey of the variational approach to discrete mechanics
and to mechanical integrators. This point of view is not confined to con-
servative systems, but also applies to forced and dissipative systems, so is
useful for control problems (for instance) as well as traditional conservative
problems in mechanics. As we shall show, the variational approach gives a
comprehensive and unified view of much of the literature on both discrete
mechanics as well as integration methods for mechanical systems and we
view these as closely allied subjects.
Some of the important topics that come out naturally from this method
are symplectic–energy-momentum methods, error analysis, constraints, forc-
ing, and Newmark algorithms. Besides giving an account of methods such
as these, we connect these techniques to other recent and exciting develop-
ments, including the PDE setting of multisymplectic spacetime integrators
(also called AVI, or asynchronous variational integrators), which are be-
ing used for problems such as nonlinear wave equations and nonlinear shell
dynamics. In fact, one of our points is that by basing the integrators on fun-
damental mechanical concepts and methods from the outset, one eases the
way to other areas, such as continuum mechanics and systems with forcing
and constraints.
In the last few years this subject has grown to be very large and an active
area of research, with many points of view and many topics. We shall focus
here on our own point of view, namely the variational view. Naturally we
must omit a number of important topics, but include several of our own.
We do make contact with some, but not all, of other topics in the final part
of this article and in the brief history below.
As in standard mechanics, some things are easier from a Hamiltonian
perspective and others are easier from a Lagrangian perspective. Regarding
symplectic integrators from both viewpoints gives greater insight into their
properties and derivations. We have tried to give a balanced perspective in
this article.
We will assume that the configuration manifold is finite-dimensional. This
means that at the outset, we will deal with the context of ordinary differ-
ential equations. However, as we have indicated, our approach is closely
tied with the variational spacetime multisymplectic approach, which is the
approach that is suitable for the infinite-dimensional, PDE context, so an
investment in the methodology of this article eases the transition to the
corresponding PDE context.

360 J. E. Marsden and M. West
One of the simple, but important ideas in discrete mechanics is easiest
to say from the Lagrangian point of view. Namely, consider a mechanical
system with configuration manifold Q. The velocity phase space is then
T Q and the Lagrangian is a map L : T Q → R. In discrete mechanics, the
starting point is to replace T Q with Q × Q and we regard, intuitively, two
nearby points as being the discrete analogue of a velocity vector.
There is an important note about constraints that we would like to say at
the outset. Recall from basic geometric mechanics (as in Marsden and Ratiu
(1999) for instance) that specifying a constraint manifold Q means that one
may already have specified constraints: for example, Q may already be a
submanifold of a linear space that is specified by constraints. However, when
constructing integrators in Section 2.1 we will take Q to be linear, although
this is only for simplicity. One way of handling a nonlinear Q is to embed it
within a linear space and use the theory of constrained systems: this point
of view is developed in Section 3. This approach has computational advan-
tages, but we will also discuss implementations of variational integrators on
arbitrary configuration manifolds Q.
1.1.1. History and literature
Of course, the variational view of mechanics goes back to Euler, Lagrange
and Hamilton. The form of the variational principle most important for con-
tinuous mechanics we use in this article is due, of course, to Hamilton (1834).
We refer to Marsden and Ratiu (1999) for additional history, references and
background on geometric mechanics.
There have been many attempts at the development of a discrete mechan-
ics and corresponding integrators that we will not attempt to survey in any
systematic fashion. The theory of discrete variational mechanics in the form
we shall use it (that uses Q×Q for the discrete analogue of the velocity phase
space) has its roots in the optimal control literature of the 1960s: see, for ex-
ample, Jordan and Polak (1964), Hwang and Fan (1967) and Cadzow (1970).
In the context of mechanics early work was done, often independently, by
Cadzow (1973), Logan (1973), Maeda (1980, 1981a, 1981b), and Lee (1983,
1987), by which point the discrete action sum, the discrete Euler–Lagrange
equations and the discrete Noether’s theorem were clearly understood. This
theory was then pursued further in the context of integrable systems in
Veselov (1988, 1991) and Moser and Veselov (1991), and in the context of
quantum mechanics in Jaroszkiewicz and Norton (1997a, 1997b) and Norton
and Jaroszkiewicz (1998).
The variational view of discrete mechanics and its numerical implemen-
tation is further developed in Wendlandt and Marsden (1997a) and (1997b)
and then extended in Kane, Marsden and Ortiz (1999a), Marsden, Pekarsky
and Shkoller (1999a, 1999b), Bobenko and Suris (1999a, 1999b) and Kane,

Discrete mechanics and variational integrators 361
Marsden, Ortiz and West (2000). The beginnings of an extension of these
ideas to a nonsmooth framework is given in Kane, Repetto, Ortiz and Mars-
den (1999b), and is carried further in Fetecau, Marsden, Ortiz and West
(2001).
Other discretizations of Hamilton’s principle are given in Mutze (1998),
Cano and Lewis (1998) and Shibberu (1994). Other versions of discrete me-
chanics (not necessarily discrete Hamilton’s principles) are given in (for in-
stance) Itoh and Abe (1988), Labudde and Greenspan (1974, 1976a, 1976b),
and MacKay (1992).
Of course, there have been many works on symplectic integration, largely
done from other points of view than that developed here. We will not at-
tempt to survey this in any systematic fashion, as the literature is simply
too large with too many points of view and too many intricate subtleties.
We give a few highlights and give further references in the body of the pa-
per. For instance, we shall connect the variational view with the generating
function point of view that was begun in De Vogela´ere (1956). Generating
function methods were developed and used in, for example, Ruth (1983),
Forest and Ruth (1990) and in Channell and Scovel (1990). See also Berg,
Warnock, Ruth and Forest (1994), and Warnock and Ruth (1991, 1992).
For an overview of symplectic integration, see Sanz-Serna (1992b) and Sanz-
Serna and Calvo (1994). Qualitative properties of symplectic integration
of Hamiltonian systems are given in Gonzalez, Higham and Stuart (1999)
and Cano and Sanz-Serna (1997). Long-time energy behaviour for oscilla-
tory systems is studied in Hairer and Lubich (2000). Long-time behaviour
of symplectic methods for systems with dissipation is given in Hairer and
Lubich (1999). A numerical study of preservation of adiabatic invariants is
given in Reich (1999b) and Shimada and Yoshida (1996). Backward error
analysis is studied in Benettin and Giorgilli (1994), Hairer (1994), Hairer
and Lubich (1997) and Reich (1999a). Other ideas connected to the above
literature include those of Baez and Gilliam (1994), Gilliam (1996), Gillilan
and Wilson (1992). For other references see the large literature on symplec-
tic methods in molecular dynamics, such as Schlick, Skeel et al. (1999), and
for various applications, see Hardy, Okunbor and Skeel (1999), Leimkuhler
and Skeel (1994), Barth and Leimkuhler (1996) and references therein.
A single-step variational idea that is relevant for our approach is given in
Ortiz and Stainier (1998), and developed further in Radovitzky and Ortiz
(1999), and Kane et al. (1999b, 2000).
Direct discretizations on the Hamiltonian side, where one discretizes the
Hamiltonian and the symplectic structure, are developed in Gonzalez (1996b)
and (1996a) and further in Gonzalez (1999) and Gonzalez et al. (1999). This
is developed and generalized much further in McLachlan, Quispel and Ro-
bidoux (1998) and (1999).

Frequently asked questions

Q1. What have the authors contributed in "Discrete mechanics and variational integrators" ?

This paper gives a review of integration algorithms for finite dimensional mechanical systems that are based on discrete variational principles. 

Q2. What are the main topics that come out naturally from this method?

Some of the important topics that come out naturally from this method are symplectic–energy-momentum methods, error analysis, constraints, forcing, and Newmark algorithms. 

Q3. What is the equivariance of the Lagrangian momentum map?

If the lifted action ΦTQ : G × TQ → TQ acts by special canonical transformations, then the Lagrangian momentum map JL : TQ→ g ∗ is equivariant. 

Q4. What are some of the problems which require nonsmooth models?

Although the authors have concentrated on mechanical systems which follow smooth trajectories, there are many physical situations which demand nonsmooth models, such as collision and fragmentation problems. 

Q5. What is the recursive rule for calculating the sequence qk N?

If the authors take initial conditions (q0, q1) then the discrete Euler–Lagrange equations define a recursive rule for calculating the sequence {qk} N k=0. 

Q6. What is the variational principle used in Section 3.1 to add forcing?

The variational principle which gives the correct equations of motion in this case is the Lagrange–d’Alembert principle used in Section 3.1 to add forcing (Bloch, Krishnaprasad, Marsden and Murray 1996a). 

Q7. What is the corresponding discrete Hamiltonian map FLd?

As the authors have seen in Section 1.6, if the discrete Lagrangian is equal to the action, then the corresponding discrete Hamiltonian map F̃Ld will exactly equal the flow FH . 

Q8. What is the simplest way to prove that the extended Lagrangian momentum map is e?

Note that in the language of multisymplectic mechanics the authors have required that the Lagrangian density be invariant, and the extended Lagrangian momentum map defined above is the fully covariant Lagrangian momentum map. 

Q9. What is the cost of recompute Aij?

While the Newton’s method outlined above typically experiences very fast convergence, it is also expensive to have to recompute Aij at each iteration of the method. 

Q10. How is the canonical oneand two-forms and Hamiltonian momentum maps related?

Using the discrete fibre derivatives it can be seen that the canonical oneand two-forms and Hamiltonian momentum maps are related to the discrete Lagrangian forms and discrete momentum maps by pullback, so thatΘ±Ld = (F ±Ld) ∗Θ, ΩLd = (F ±Ld) ∗Ω, and J±Ld = (F ±Ld) ∗JH . 

Q11. What is the condition for L1d to be a null discrete Lagrangian?

A sufficient (and presumably necessary) condition for this to be true is that their difference L∆d = L 1 d − L 2 d is a null discrete Lagrangian; that is, the discrete Euler–Lagrange equations for L∆d are satisfied by any triplet (q0, q1, q2).